How to prove that $(\Bbb{Z}[t]+t\Bbb{R}[t])/t\Bbb{R}[t]\cong\Bbb{Z}\cong\Bbb{Z}[t]/t\Bbb{Z}[t]\cong\Bbb{Z}[t]/(\Bbb{Z}[t]\cap t\Bbb{R}[t])$?

Question

I already proved that $(\mathbb{Z}[t]+t\mathbb{R}[t])/t\mathbb{R}[t]\cong\mathbb{Z}[t]/(\mathbb{Z}[t]\cap t\mathbb{R}[t])$ with the first isomorphism theorem but i do not know how to continue.

Answer 1

The equality $\mathbb{Z}[t]\cap t\mathbb{Z}[t]=t\mathbb{Z}[t]$ is clear (if you're not convinced, take 30sec to prove it), so the last isomorphism is in fact an equality.

It remains to prove that $\mathbb{Z}[t]/t\mathbb{Z}[t]\simeq\mathbb{Z}$, which can be achieved by applying the first isomorphism theorem to evaluation at $0$.